Simplify 1/(1+tan(x)^2)+1/(1+cot(x)^2)

$\frac{1}{1 + \tan^{2} (x)} + \frac{1}{1 + \cot^{2} (x)}$

Step 1

Rearrange terms.

$\frac{1}{\tan^{2} (x) + 1} + \frac{1}{1 + \cot^{2} (x)}$

Step 2

Apply pythagorean identity.

$\frac{1}{\sec^{2} (x)} + \frac{1}{1 + \cot^{2} (x)}$

Step 3

Apply pythagorean identity.

$\frac{1}{\sec^{2} (x)} + \frac{1}{\csc^{2} (x)}$

Step 4

Simplify each term.

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2}}{\sec^{2} (x)} + \frac{1}{\csc^{2} (x)}$

Rewrite $\frac{1^{2}}{\sec^{2} (x)}$ as ${(\frac{1}{\sec (x)})}^{2}$ .

${(\frac{1}{\sec (x)})}^{2} + \frac{1}{\csc^{2} (x)}$

Rewrite $\sec (x)$ in terms of sines and cosines.

${(\frac{1}{\frac{1}{\cos (x)}})}^{2} + \frac{1}{\csc^{2} (x)}$

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

${(1 \cos (x))}^{2} + \frac{1}{\csc^{2} (x)}$

Multiply $\cos (x)$ by $1$ .

$\cos^{2} (x) + \frac{1}{\csc^{2} (x)}$

Rewrite $1$ as $1^{2}$ .

$\cos^{2} (x) + \frac{1^{2}}{\csc^{2} (x)}$

Rewrite $\frac{1^{2}}{\csc^{2} (x)}$ as ${(\frac{1}{\csc (x)})}^{2}$ .

$\cos^{2} (x) + {(\frac{1}{\csc (x)})}^{2}$

Rewrite $\csc (x)$ in terms of sines and cosines.

$\cos^{2} (x) + {(\frac{1}{\frac{1}{\sin (x)}})}^{2}$

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\cos^{2} (x) + {(1 \sin (x))}^{2}$

Multiply $\sin (x)$ by $1$ .

$\cos^{2} (x) + \sin^{2} (x)$

Step 5

Rearrange terms.

$\sin^{2} (x) + \cos^{2} (x)$

Step 6

Apply pythagorean identity.

$1$